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On the complexity of monotone circuits for threshold symmetric Boolean functions 关于阈值对称布尔函数单调电路的复杂性
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-10-01 DOI: 10.1515/dma-2021-0031
I. Sergeev
{"title":"On the complexity of monotone circuits for threshold symmetric Boolean functions","authors":"I. Sergeev","doi":"10.1515/dma-2021-0031","DOIUrl":"https://doi.org/10.1515/dma-2021-0031","url":null,"abstract":"Abstract The complexity of implementation of a threshold symmetric n-place Boolean function with threshold k = O(1) via circuits over the basis {∨, ∧} is shown not to exceed 2 log2 k ⋅ n + o(n). Moreover, the complexity of a threshold-2 function is proved to be 2n + Θ( n $begin{array}{} sqrt n end{array} $), and the complexity of a threshold-3 function is shown to be 3n + O(log n), the corresponding lower bounds are put forward.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"345 - 366"},"PeriodicalIF":0.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43093321","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Boolean analogues of the Pascal triangle with maximal possible number of ones 具有最大可能数目的帕斯卡三角形的布尔类比
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-10-01 DOI: 10.1515/dma-2021-0029
F. M. Malyshev
{"title":"Boolean analogues of the Pascal triangle with maximal possible number of ones","authors":"F. M. Malyshev","doi":"10.1515/dma-2021-0029","DOIUrl":"https://doi.org/10.1515/dma-2021-0029","url":null,"abstract":"Abstract The aim of the paper is to find the maximal possible number ξ of units in Boolean triangular array Ts formed by s(s+1)2 $begin{array}{} displaystyle frac{s(s+1)}{2} end{array}$ elements of the field GF(2) defined by the top row of s elements. Each element of each row except the top one is the sum (as in the Pascal’s triangle) of two elements of the above row. It is proved that ξ ⩽ ⌈ s(s+1)3 $begin{array}{} displaystyle frac{s(s+1)}{3} end{array}$⌉ and this value is attained only on triangles having the upper row as the Fibonacci series mod 2.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"319 - 325"},"PeriodicalIF":0.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47623873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The limited deficit method and the problem of constructing orthomorphisms and almost orthomorphisms of Abelian groups 有限亏缺法与构造阿贝尔群的正态和几乎正态问题
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-10-01 DOI: 10.1515/dma-2021-0030
A. V. Menyachikhin
{"title":"The limited deficit method and the problem of constructing orthomorphisms and almost orthomorphisms of Abelian groups","authors":"A. V. Menyachikhin","doi":"10.1515/dma-2021-0030","DOIUrl":"https://doi.org/10.1515/dma-2021-0030","url":null,"abstract":"Abstract The limited deficit method is described, which allows constructing new orthomorphisms (almost orthomorphisms) of groups with the use of those already known. A class of transformations is described under which the set of all orthomorphisms (almost orthomorphisms) remains invariant. It is conjectured that the set of all orthomorphisms (almost orthomorphisms) is generated by transformations implemented by the limited deficit method. This conjecture is verified for all Abelian groups of order at most 12. The spectral-linear method and the spectral-differential method of design of permutations over the additive group of the field 𝔽2m (m = 4, …, 8) are used to construct orthomorphisms with sufficiently high values of the most important cryptographic parameters.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"327 - 343"},"PeriodicalIF":0.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46061616","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the average-case complexity of Boolean functions under binomial distribution on their domains 二项分布布尔函数在其定义域上的平均情况复杂度
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-10-01 DOI: 10.1515/dma-2021-0028
A. V. Chashkin
{"title":"On the average-case complexity of Boolean functions under binomial distribution on their domains","authors":"A. V. Chashkin","doi":"10.1515/dma-2021-0028","DOIUrl":"https://doi.org/10.1515/dma-2021-0028","url":null,"abstract":"Abstract Given a binomial probability distribution on the n-dimensional Boolean cube, the complexity of implementation of Boolean functions by straight line programs with conditional stop is considered. The order, as n → ∞, of the average-case complexity of almost all n-place Boolean functions is established.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"315 - 318"},"PeriodicalIF":0.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42189872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Large deviations of branching process in a random environment 随机环境中分支过程的大偏差
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-08-01 DOI: 10.1515/dma-2021-0025
A. V. Shklyaev
{"title":"Large deviations of branching process in a random environment","authors":"A. V. Shklyaev","doi":"10.1515/dma-2021-0025","DOIUrl":"https://doi.org/10.1515/dma-2021-0025","url":null,"abstract":"Abstract In this first part of the paper we find the asymptotic formulas for the probabilities of large deviations of the sequence defined by the random difference equation Yn+1=AnYn + Bn, where A1, A2, … are independent identically distributed random variables and Bn may depend on {(Ak,Bk),0⩽k<n} $ {(A_k,B_k),0leqslant k lt n} $ for any n≥1. In the second part of the paper this results are applied to the large deviations of branching processes in a random environment.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"281 - 291"},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44535506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
On closed classes in partial k-valued logic that contain all polynomials 关于包含所有多项式的部分k值逻辑中的闭类
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-08-01 DOI: 10.1515/dma-2021-0020
V. Alekseev
{"title":"On closed classes in partial k-valued logic that contain all polynomials","authors":"V. Alekseev","doi":"10.1515/dma-2021-0020","DOIUrl":"https://doi.org/10.1515/dma-2021-0020","url":null,"abstract":"Abstract Let Polk be the set of all functions of k-valued logic representable by a polynomial modulo k, and let Int (Polk) be the family of all closed classes (with respect to superposition) in the partial k-valued logic containing Polk and consisting only of functions extendable to some function from Polk. Previously the author showed that if k is the product of two different primes, then the family Int (Polk) consists of 7 closed classes. In this paper, it is proved that if k has at least 3 different prime divisors, then the family Int (Polk) contains an infinitely decreasing (with respect to inclusion) chain of different closed classes.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"231 - 240"},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42312171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Collisions and incidence of vertices and components in the graph of k-fold iteration of the uniform random mapping 图中k次迭代的一致随机映射的顶点和分量的碰撞和关联
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-08-01 DOI: 10.1515/dma-2021-0023
V. O. Mironkin
{"title":"Collisions and incidence of vertices and components in the graph of k-fold iteration of the uniform random mapping","authors":"V. O. Mironkin","doi":"10.1515/dma-2021-0023","DOIUrl":"https://doi.org/10.1515/dma-2021-0023","url":null,"abstract":"Abstract The probabilistic characteristics of the graph of k-fold iteration of uniform random mapping are studied. Formulas for the distribution of the length of the aperiodicity segment of an arbitrary vertex with some restrictions are calculated. We obtain exact expressions for the probabilities that two arbitrary vertices belong to the same connected component, that an arbitrary vertex belongs to the preimage set of another vertex and that there exists a collision in the considered graph.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"259 - 269"},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48654805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Alphabetic points in compositions and words 作文和单词中的字母点
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-08-01 DOI: 10.1515/dma-2021-0021
M. Archibald, A. Blecher, A. Knopfmacher
{"title":"Alphabetic points in compositions and words","authors":"M. Archibald, A. Blecher, A. Knopfmacher","doi":"10.1515/dma-2021-0021","DOIUrl":"https://doi.org/10.1515/dma-2021-0021","url":null,"abstract":"Abstract We use generating functions to account for alphabetic points (or the lack thereof) in compositions and words. An alphabetic point is a value j such that all the values to its left are not larger than j and all the values to its right are not smaller than j. We also provide the asymptotics for compositions and words which have no alphabetic points, as the size tends to infinity. This is achieved by the construction of upper and lower bounds which converge to each other, and in the latter case by probabilistic arguments.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"241 - 250"},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45452622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Frontmatter Frontmatter
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-08-01 DOI: 10.1515/dma-2021-frontmatter4
{"title":"Frontmatter","authors":"","doi":"10.1515/dma-2021-frontmatter4","DOIUrl":"https://doi.org/10.1515/dma-2021-frontmatter4","url":null,"abstract":"","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47119490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Local limit theorems for generalized scheme of allocation of particles into ordered cells 广义粒子有序单元分配方案的局部极限定理
IF 0.5
Discrete Mathematics and Applications Pub Date : 2021-08-01 DOI: 10.1515/dma-2021-0026
A. N. Timashev
{"title":"Local limit theorems for generalized scheme of allocation of particles into ordered cells","authors":"A. N. Timashev","doi":"10.1515/dma-2021-0026","DOIUrl":"https://doi.org/10.1515/dma-2021-0026","url":null,"abstract":"Abstract A generalized scheme of allocation of n particles into ordered cells (components). Some statements containing sufficient conditions for the weak convergence of the number of components with given cardinality and of the total number of components to the negative binomial distribution as n → ∞ are presented as hypotheses. Examples supporting the validity of these statements in particular cases are considered. For some examples we prove local limit theorems for the total number of components which partially generalize known results on the convergence of this distribution to the normal law.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"293 - 307"},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46045268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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